Neural Fields, Finite-Dimensional Approximation, Large Deviations, - - PowerPoint PPT Presentation

Neural Fields, Finite-Dimensional Approximation, Large Deviations, and SDE Continuation Christian Kuehn

Vienna University of Technology

Outline

Part 1: Neural Fields (joint work with Martin Riedler, Linz/Vienna):

• 1. Neural Fields - Amari-type
• 2. Galerkin Approximation
• 3. Large Deviation Principle(s)

Part 2: SDE Continuation

• 1. Numerical Continuation
• 2. Extension to SODEs
• 3. Calculating Kramers’ Law
• 4. Extension to SPDEs

Neural Fields

Amari-type neural field model: dUt(x) =

• −αUt(x) +
• B

w(x, y)f (Ut(y)) dy

• dt + ε dWt(x).

Neural Fields

Amari-type neural field model: dUt(x) =

• −αUt(x) +
• B

w(x, y)f (Ut(y)) dy

• dt + ε dWt(x).

Ingredients:

◮ B ⊂ Rd bounded closed domain. Hilbert space X = L2(B). ◮ (x, t) ∈ B × [0, T], u = u(x, t) ∈ R, α > 0, 0 < ε ≪ 1.

Neural Fields

Amari-type neural field model: dUt(x) =

• −αUt(x) +
• B

w(x, y)f (Ut(y)) dy

• dt + ε dWt(x).

Ingredients:

◮ B ⊂ Rd bounded closed domain. Hilbert space X = L2(B). ◮ (x, t) ∈ B × [0, T], u = u(x, t) ∈ R, α > 0, 0 < ε ≪ 1. ◮ w : B × B → R kernel, modelling neural connectivity. ◮ f : R → (0, +∞) gain function, modelling neural input.

Neural Fields

Amari-type neural field model: dUt(x) =

• −αUt(x) +
• B

w(x, y)f (Ut(y)) dy

• dt + ε dWt(x).

Ingredients:

◮ B ⊂ Rd bounded closed domain. Hilbert space X = L2(B). ◮ (x, t) ∈ B × [0, T], u = u(x, t) ∈ R, α > 0, 0 < ε ≪ 1. ◮ w : B × B → R kernel, modelling neural connectivity. ◮ f : R → (0, +∞) gain function, modelling neural input. ◮ Q : X → X trace-class, non-negative symmetric operator:

eigenvalues λ2

i ∈ R, eigenfunctions vi. ◮ Wt(x) := ∞ i=1 λiβi tvi(x),

βi

t iid Brownian motions.

Existence and Regularity

Assumptions: ◮ Kg(x) :=

B w(x, y)g(y) dy is a compact self-adjoint operator on L2(B).

◮ F(g)(x) := f (g(x)) is a Lipschitz continuous Nemytzkii operator on L2(B). Neural field as evolution equation dUt = [−αUt + KF(Ut)] dt + ε dWt.

Existence and Regularity

Assumptions: ◮ Kg(x) :=

B w(x, y)g(y) dy is a compact self-adjoint operator on L2(B).

◮ F(g)(x) := f (g(x)) is a Lipschitz continuous Nemytzkii operator on L2(B). Neural field as evolution equation dUt = [−αUt + KF(Ut)] dt + ε dWt. (daPrato-Zabczyk92) ⇒ Mild solution u ∈ C([0, T], L2(B)) Ut = e−αtU0 + t e−α(t−s)KF(Us) ds + ε t e−α(t−s) dWs.

Existence and Regularity

Assumptions: ◮ Kg(x) :=

B w(x, y)g(y) dy is a compact self-adjoint operator on L2(B).

◮ F(g)(x) := f (g(x)) is a Lipschitz continuous Nemytzkii operator on L2(B). Neural field as evolution equation dUt = [−αUt + KF(Ut)] dt + ε dWt. (daPrato-Zabczyk92) ⇒ Mild solution u ∈ C([0, T], L2(B)) Ut = e−αtU0 + t e−α(t−s)KF(Us) ds + ε t e−α(t−s) dWs.

Lemma (K./Riedler, 2013)

vi Lipschitz with constants Li and for some ρ ∈ (0, 1) sup

x∈B

• ∞
• i=1

λ2

i vi(x)2

• < ∞ ,

sup

x∈B

• ∞
• i=1

λ2

i L2ρ i |vi(x)|2(1−ρ)

• < ∞

⇒ u ∈ C([0, T], C(B)).

Galerkin Approximation

Spectral representation of solution: Ut(x) =

∞

• i=1

ui

t vi(x) .

Galerkin Approximation

Spectral representation of solution: Ut(x) =

∞

• i=1

ui

t vi(x) .

Take L2-inner product with vi in neural field model dUt, vi =

• −αUt, vi + KF(Ut), vi
• dt + εdWt, vi,

⇒ dui

t

=

• −αui

t + (KF)i(u1 t , u2 t , . . .)

• dt + ελi dβi

t.

Galerkin Approximation

Spectral representation of solution: Ut(x) =

∞

• i=1

ui

t vi(x) .

Take L2-inner product with vi in neural field model dUt, vi =

• −αUt, vi + KF(Ut), vi
• dt + εdWt, vi,

⇒ dui

t

=

• −αui

t + (KF)i(u1 t , u2 t , . . .)

• dt + ελi dβi

t.

where (KF)i(u1

t , u2 t , . . .) :=

• B

f  

∞

• j=1

uj

t vj(x)

 

• B

w(x, y)vi(y)dy

• dx

Approximation Accuracy

Theorem (K./Riedler, 2013)

For all T > 0 lim

N→∞ supt∈[0,T] Ut − UN t L2(B) = 0

a.s.

Approximation Accuracy

Theorem (K./Riedler, 2013)

For all T > 0 lim

N→∞ supt∈[0,T] Ut − UN t L2(B) = 0

a.s. If “regularity-lemma” conditions hold and U0 ∈ C(B) such that lim

N→∞ U0 − PNU0C(B) = 0

then lim

N→∞ supt∈[0,T] Ut − UN t C(B) = 0

a.s.

Proof.

Lengthy calculation using a technique by Bl¨

• mker/Jentzen

(SINUM 2013).

Large Deviations Principle (LDP)

Example: Stochastic ordinary differential equation dut = g(ut) dt + εG(ut) dβt. where

◮ ut ∈ RN, g : RN → RN, G : RN → RN×k, ◮ βt = (β1 t , . . . , βk t )T vector of k iid Brownian motions, ◮ u0 ∈ D ⊂ RN.

Large Deviations Principle (LDP)

Example: Stochastic ordinary differential equation dut = g(ut) dt + εG(ut) dβt. where

◮ ut ∈ RN, g : RN → RN, G : RN → RN×k, ◮ βt = (β1 t , . . . , βk t )T vector of k iid Brownian motions, ◮ u0 ∈ D ⊂ RN.

Goal: Estimate first-exit time τ ε

D := inf{t > 0 : ut = uε t ∈ D}.

An Abstract Theorem

◮ X := C0([0, T], RN) = {φ ∈ C([0, T], RN) : φ(0) = u0}. ◮ HN

1 := {φ : [0, T] → RN : φ absolutely continuous, φ′ ∈ L2, φ(0) = 0}.

◮ Diffusion matrix D(u) := G(u)T G(u) ∈ RN×N positive definite.

An Abstract Theorem

◮ X := C0([0, T], RN) = {φ ∈ C([0, T], RN) : φ(0) = u0}. ◮ HN

1 := {φ : [0, T] → RN : φ absolutely continuous, φ′ ∈ L2, φ(0) = 0}.

◮ Diffusion matrix D(u) := G(u)T G(u) ∈ RN×N positive definite.

Theorem (Freidlin, Wentzell)

The SODE satisfies an LDP − inf

Γo I

≤ lim inf

ε→0 ε2 ln P((uε t )t∈[0,T] ∈ Γ) ≤

≤ lim sup

ε→0

ε2 ln P((uε

t )t∈[0,T] ∈ Γ) ≤

− inf

¯ Γ

I. for any measurable set of paths Γ ⊂ X with rate function I(φ) = 1

2

T

0 (φ′ t − g(φt))T D(φt)−1(φ′ t − g(φt))dt,

φ ∈ u0 + HN

1 ,

+∞

• therwise.

Arhennius-Eyring-Kramers’ Formula

◮ Gradient structure and additive noise

dut = −∇V (ut) dt + εId dβt.

◮ V has precisely two local minima u∗

±, single saddle point u∗ s .

◮ Hessian ∇2V (u∗

s ) at saddle has eigenvalues

ρ1(u∗

s ) < 0 < ρ2(u∗ s ) < · · · < ρN(u∗ s ).

Arhennius-Eyring-Kramers’ Formula

◮ Gradient structure and additive noise

dut = −∇V (ut) dt + εId dβt.

◮ V has precisely two local minima u∗

±, single saddle point u∗ s .

◮ Hessian ∇2V (u∗

s ) at saddle has eigenvalues

ρ1(u∗

s ) < 0 < ρ2(u∗ s ) < · · · < ρN(u∗ s ).

Theorem (Kramers’ Formula)

Mean first-passage u∗

− to u∗ + obeys:

E[τu∗

−→u∗ +}] ∼

2π |ρ1(u∗

s )|

• | det(∇2V (u∗

s ))|

det(∇2V (u∗

−)) e2(V (u∗

s )−V (u∗ −))/ε2.

Back to Neural Fields... Kramers’ Formula and LDP

Observations (K./Riedler, 2013)

◮ From [Laing/Troy03,Enulescu/Bestehorn07] ε = 0 ⇒ neural

field has energy-structure. Let g := f −1, P(x, t) = f (U(x, t)). ∂tP(x, t) = − 1 g′(P(x, t))∇E[P(x, t)].

Back to Neural Fields... Kramers’ Formula and LDP

Observations (K./Riedler, 2013)

◮ From [Laing/Troy03,Enulescu/Bestehorn07] ε = 0 ⇒ neural

field has energy-structure. Let g := f −1, P(x, t) = f (U(x, t)). ∂tP(x, t) = − 1 g′(P(x, t))∇E[P(x, t)]. But, there are problems for ε > 0 ⇒

◮ Change-of-variable ⇒ multiplicative noise. ◮ Space-time dependent factor 1/g ′(P(x, t)). ◮ Trace-class noise.

Back to Neural Fields... Kramers’ Formula and LDP

Observations (K./Riedler, 2013)

◮ From [Laing/Troy03,Enulescu/Bestehorn07] ε = 0 ⇒ neural

field has energy-structure. Let g := f −1, P(x, t) = f (U(x, t)). ∂tP(x, t) = − 1 g′(P(x, t))∇E[P(x, t)]. But, there are problems for ε > 0 ⇒

◮ Change-of-variable ⇒ multiplicative noise. ◮ Space-time dependent factor 1/g ′(P(x, t)). ◮ Trace-class noise.

◮ LDP follows from evolution equation [daPratoZabczyk92]. ◮ LDP can be approximated using Galerkin method.

Part 2 SDE Continuation: Motivation

Consider the general differential equation ∂u ∂t = F(u; λ) where λ ∈ Rp are parameters.

Part 2 SDE Continuation: Motivation

Consider the general differential equation ∂u ∂t = F(u; λ) where λ ∈ Rp are parameters. F(u; λ) could lead to ODE, DDE, PDE, SDE, SPDE, etc. Problem: Forward simulation is usually very restrictive!

• 1. Simulate over initial values u0.
• 2. Simulate over parameter space µ ∈ Rp.
• 3. Simulate over noise realizations ω ∈ Ω.

Do you really understand the nonlinear dynamics from averages?

Deterministic DEs Standard Method: Continuation

Consider the ODE x′ = f (x; µ), f : Rn × R → Rn. Let (x; µ) =: y. A curve y = γ(s) of equilibria satisfies f (γ(s)) = 0. (note: Df (γ(0))γ′(0) = 0) y0 := γ(0) ¯ y1 f (y) = 0

(a) Prediction Step

y1 ¯ y1 f (y) = 0

(b) Correction Step

Important: Excellent guess from (a) for Newton’s Method in (b).

Numerical Bifurcation Analysis for Stochastic Systems?

Consider the stochastic (ordinary) differential equation (SDE) dxt = g(xt; µ) dt + εG(xt; µ) dWt, xt ∈ Rn, Wt = (W1,t, W2,t, . . . , Wk,t)T Brownian motion, F(xt; µ) ∈ Rn×k; let D(x; µ) := G(x; µ)G(x; µ)T .

Numerical Bifurcation Analysis for Stochastic Systems?

Consider the stochastic (ordinary) differential equation (SDE) dxt = g(xt; µ) dt + εG(xt; µ) dWt, xt ∈ Rn, Wt = (W1,t, W2,t, . . . , Wk,t)T Brownian motion, F(xt; µ) ∈ Rn×k; let D(x; µ) := G(x; µ)G(x; µ)T .

◮ Appproach 1: Forward Monte-Carlo simulation. ◮ Problems: Sampling often prohibitive.

Numerical Bifurcation Analysis for Stochastic Systems?

Consider the stochastic (ordinary) differential equation (SDE) dxt = g(xt; µ) dt + εG(xt; µ) dWt, xt ∈ Rn, Wt = (W1,t, W2,t, . . . , Wk,t)T Brownian motion, F(xt; µ) ∈ Rn×k; let D(x; µ) := G(x; µ)G(x; µ)T .

◮ Appproach 1: Forward Monte-Carlo simulation. ◮ Problems: Sampling often prohibitive. ◮ Appproach 2: Use probability density p = p(x, t). Requires

Fokker-Planck solution ∂p ∂t = −

n

• i=1

∂ ∂xi (g(x; µ)p) + ε2 2

n

• i,j=1

∂ ∂xi∂xj (Dij(x; µ)p).

◮ Problems: High-dimensional PDE; not even ε = 0 is good!

Strategy - Generalization to SDEs

Step 1: Recall dxt = g(xt; µ) dt + εG(xt; µ) dWt. Step 2: Expand near (locally stable) deterministic equilibrium x∗ dXt = A(x∗; µ)Xt dt + εF(x∗; µ) dWt where A(x; µ) = (Dxf )(x; µ) ∈ Rn×n.

Strategy - Generalization to SDEs

Step 1: Recall dxt = g(xt; µ) dt + εG(xt; µ) dWt. Step 2: Expand near (locally stable) deterministic equilibrium x∗ dXt = A(x∗; µ)Xt dt + εF(x∗; µ) dWt where A(x; µ) = (Dxf )(x; µ) ∈ Rn×n. Step 3: The covariance matrix Ct := Cov(Xt) solves C ′

t

= A(x∗; µ)Ct + CtA(x∗; µ)T + ε2G(x∗; µ)G(x∗; µ)T

• equil. ⇒ 0

= A(x∗; µ)C + CA(x∗; µ)T + ε2G(x∗; µ)G(x∗; µ)T

Strategy - Generalization to SDEs

Step 1: Recall dxt = g(xt; µ) dt + εG(xt; µ) dWt. Step 2: Expand near (locally stable) deterministic equilibrium x∗ dXt = A(x∗; µ)Xt dt + εF(x∗; µ) dWt where A(x; µ) = (Dxf )(x; µ) ∈ Rn×n. Step 3: The covariance matrix Ct := Cov(Xt) solves C ′

t

= A(x∗; µ)Ct + CtA(x∗; µ)T + ε2G(x∗; µ)G(x∗; µ)T

• equil. ⇒ 0

= A(x∗; µ)C + CA(x∗; µ)T + ε2G(x∗; µ)G(x∗; µ)T Step 4: Define the covariance ellipsoid B(h) :=

• x ∈ Rn : (x − x∗)TC −1(x − x∗) ≤ h2

.

Covariance Ellipsoids via Continuation

Important observations:

◮ Continue the equilibrium x∗ = x∗(µ) as usual. ◮ For covariance ellipsoid one has to solve a Lyapunov equation

AC + CAT + B = 0

◮ During continuation the matrix

Dxg(x∗; µ) = A(x∗; µ) = A is available as a submatrix of Dg(x∗; µ).

Covariance Ellipsoids via Continuation

Important observations:

◮ Continue the equilibrium x∗ = x∗(µ) as usual. ◮ For covariance ellipsoid one has to solve a Lyapunov equation

AC + CAT + B = 0

◮ During continuation the matrix

Dxg(x∗; µ) = A(x∗; µ) = A is available as a submatrix of Dg(x∗; µ).

◮ Efficient iterative methods for Lyapunov equations exist. ◮ A simple initial guess for C(µ2) at (x∗(µ2), µ2) is

C(x∗(µ1); µ1).

Ellipsoids and Distance

Question: What is the distance between ellipsoids?

Ellipsoids and Distance

Question: What is the distance between ellipsoids? Let Q be positive semi-definite then E :=

• x ∈ Rn : v Tx ≤ v Tx∗ + (v T Qv)1/2

∀v ∈ Rn . defines an ellipsoid centered at x∗.

Ellipsoids and Distance

Question: What is the distance between ellipsoids? Let Q be positive semi-definite then E :=

• x ∈ Rn : v Tx ≤ v Tx∗ + (v T Qv)1/2

∀v ∈ Rn . defines an ellipsoid centered at x∗. Fact: May solve an optimization problem δ = δ(E(x∗

1 , Q1), E(x∗ 2 , Q2))

= max

v=1

• v Tx∗

1 − (v T Q1v)1/2 − v Tx∗ 2 − (v TQ2v)1/2

. Idea: Use iterative method (e.g. SQP) & initial guess from continuation to compute δ.

Neural Competition

Consider two neural populations x′

1

= −x1 + S(Ic − βx2 − gy1), x′

2

= −x2 + S(Ic − βx1 − gy2), y ′

1

= ǫ(x1 − y1), y ′

2

= ǫ(x2 − y2), where

◮ x1,2 = averaged firing rates, ◮ y1,2 = fatigue/reset variables, ◮ S(u) := 1 1+exp(−r(u−θ)).

Neural Competition

Consider two neural populations x′

1

= −x1 + S(Ic − βx2 − gy1), x′

2

= −x2 + S(Ic − βx1 − gy2), y ′

1

= ǫ(x1 − y1), y ′

2

= ǫ(x2 − y2), where

◮ x1,2 = averaged firing rates, ◮ y1,2 = fatigue/reset variables, ◮ S(u) := 1 1+exp(−r(u−θ)).

Look at noisy fast subsystem ǫ = 0 dx1 dx2

• =

−x1 + S(Ic − βx2 − gy1) −x2 + S(Ic − βx1 − gy2)

• dt + ε2G(x) dWt

Numerical Continuation...

0.6 0.8 1 1.2 1.4 1.6 −0.5 0.5 1 1.5 1 2 3 −0.4 0.4 0.8 1.2 −0.4 0.4 0.8 1.2 0.6 0.8 1 1.2 1.4 1.6 −0.5 0.5 1 1.5

(a) (b) (c)

Ic Ic Ic δ x1 x1 x2 For parameter values y1 = 0.7, y2 = 0.75, β = 1.1, g = 0.5, r = 10, θ = 0.2. and ε2G(x∗)G(x∗)T = ε2

• 1

0.4 0.4 1

• for ε2 = 0.3.

Metastability and Noise-Induced Switching

Consider a gradient system dxt = −∇Vµ(xt) dt + ε dWt, Vµ : Rn → R. (1) Assume

◮ two stable equilibria x∗ and y ∗ ◮ saddle z∗, one unstable direction eigenvalue λ(z∗; µ) > 0

Kramers’ Law E[τx∗→y ∗] = 2π |λ(z∗; µ)|

• | det(A(z∗; µ))|

det(A(x∗; µ)) e2[Vµ(z∗)−Vµ(x∗)]/ε2 where A(x∗; µ) = D2Uµ(x∗; µ) ∈ Rn×n.

Continuation and Kramers’ Law

Kramers’ Law E[τx∗→y ∗] = 2π |λ(z∗; µ)|

• | det(A(z∗; µ))|

det(A(x∗; µ)) e2[Vµ(z∗)−Vµ(x∗)]/ε2 Observations:

◮ Just continue the equilibria x∗, y ∗, z∗ as usual. ◮ Jacobian A(z∗; µ) is available. ◮ Compute det(A(x∗; µ)) via LU decomposition. ◮ Leading eigenvalue λ(z∗; µ) may use Rayleigh iteration.

Extension to SPDEs

Starting point: (cubic-quintic) Allen-Cahn PDE ∂u ∂t = ∆u − 4(µu + u3 − u5). u = u(x, t), x ∈ Ω ⊂ R2, given boundary conditions.

Extension to SPDEs

Starting point: (cubic-quintic) Allen-Cahn PDE ∂u ∂t = ∆u − 4(µu + u3 − u5). u = u(x, t), x ∈ Ω ⊂ R2, given boundary conditions. Main Steps:

• 1. Compute bifurcation for PDE (e.g. → pde2path).

Extension to SPDEs

Starting point: (cubic-quintic) Allen-Cahn PDE ∂u ∂t = ∆u − 4(µu + u3 − u5) + g(u)ξ. u = u(x, t), x ∈ Ω ⊂ R2, given boundary conditions. Main Steps:

• 1. Compute bifurcation for PDE (e.g. → pde2path).
• 2. Consider the SPDE version (e.g. → trace-class noise).
• 3. Discretize in space (e.g. → FDM, FEM, Galerkin).

Extension to SPDEs

Starting point: (cubic-quintic) Allen-Cahn PDE ∂u ∂t = ∆u − 4(µu + u3 − u5) + g(u)ξ. u = u(x, t), x ∈ Ω ⊂ R2, given boundary conditions. Main Steps:

• 1. Compute bifurcation for PDE (e.g. → pde2path).
• 2. Consider the SPDE version (e.g. → trace-class noise).
• 3. Discretize in space (e.g. → FDM, FEM, Galerkin).
• 4. Apply numerical continuation for SDEs.

PDE: Deterministic Numerical Continuation

u2 µ

x x x y y y u u u

(a) (b) (c) (d)

(c) (d) (b)

R0 R1 R2 R3 R4

SPDE: Stochastic Numerical Continuation

10

−2

10

−1

10 10

−2

10

−1

10 10

1

10

2

1.2 1.4 1.6 1.8 2 −0.5 0.5 1 1.5 2

· µ µf

1 − µ

· (a) (b) ◮ scaling law of the variance near bifurcation point ◮ link to early-warning signs ◮ Computation on standard desktop computer for SPDEs

Overview

◮ Infinite-dimensional neural fields ◮ Numerical continuation methods for SODEs ◮ Numerics extends to SPDEs and SPIDEs

Overview

◮ Infinite-dimensional neural fields ◮ Numerical continuation methods for SODEs ◮ Numerics extends to SPDEs and SPIDEs

A general strategy:

• 1. Abstract stochastic analysis
• 2. Conversion into numerical deterministic problem
• 3. Continuation and iterative methods

Overview

◮ Infinite-dimensional neural fields ◮ Numerical continuation methods for SODEs ◮ Numerics extends to SPDEs and SPIDEs

A general strategy:

• 1. Abstract stochastic analysis
• 2. Conversion into numerical deterministic problem
• 3. Continuation and iterative methods

◮ see also: www.asc.tuwien.ac.at/∼ckuehn and arXiv Remark: Multiscale Dynamics (almost) everywhere!

References

• K. Gowda and C. Kuehn.

Warning signs for pattern-formation in SPDEs.

• Comm. Nonl. Sci. & Numer. Simul., 22(1):55–69, 2015.
• C. Kuehn.

A mathematical framework for critical transitions: bifurcations, fast-slow systems and stochastic dynamics. Physica D, 240(12):1020–1035, 2011.

• C. Kuehn.

Deterministic continuation of stochastic metastable equilibria via Lyapunov equations and ellipsoids. SIAM J. Sci. Comp., 34(3):A1635–A1658, 2012.

• C. Kuehn.

A mathematical framework for critical transitions: normal forms, variance and applications.

• J. Nonlinear Sci., 23(3):457–510, 2013.
• C. Kuehn.

Numerical continuation and SPDE stability for the 2d cubic-quintic Allen-Cahn equation. arXiv:1408.4000, pages 1–26, 2014.

• C. Kuehn and M.G. Riedler.

Large deviations for nonlocal stochastic neural fields.

• J. Math. Neurosci., 4(1):1–33, 2014.

References

• K. Gowda and C. Kuehn.

Warning signs for pattern-formation in SPDEs.

• Comm. Nonl. Sci. & Numer. Simul., 22(1):55–69, 2015.
• C. Kuehn.

A mathematical framework for critical transitions: bifurcations, fast-slow systems and stochastic dynamics. Physica D, 240(12):1020–1035, 2011.

• C. Kuehn.

Deterministic continuation of stochastic metastable equilibria via Lyapunov equations and ellipsoids. SIAM J. Sci. Comp., 34(3):A1635–A1658, 2012.

• C. Kuehn.

A mathematical framework for critical transitions: normal forms, variance and applications.

• J. Nonlinear Sci., 23(3):457–510, 2013.
• C. Kuehn.

Numerical continuation and SPDE stability for the 2d cubic-quintic Allen-Cahn equation. arXiv:1408.4000, pages 1–26, 2014.

• C. Kuehn and M.G. Riedler.

Large deviations for nonlocal stochastic neural fields.

• J. Math. Neurosci., 4(1):1–33, 2014.

Thank you for your attention.